
theorem ::$N Inequality of arithmetic and geometric means
  for f being non empty positive real-valued FinSequence holds
    GMean f <= Mean f
  proof
    let f be non empty positive real-valued FinSequence;
    per cases;
    suppose Het f = 0; then
      reconsider ff = f as
        non empty homogeneous positive real-valued FinSequence;
      GMean ff = Mean ff by HetBase;
      hence thesis;
    end;
    suppose Het f <> 0; then
      reconsider ff = f as
        non empty heterogeneous positive real-valued FinSequence
          by HetHetero;
      ex g being non empty homogeneous positive real-valued FinSequence st
        GMean g > GMean ff & Mean g = Mean ff by WOWTheo;
      hence thesis by HetBase;
    end;
  end;
